FERMAT S THEOREM ON BINARY POWERS

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1 NNTDM (00), - ERMAT S THEOREM ON BINARY POWERS J. V. Leyedekkes The Uivesity of Sydey, 00, Austalia A. G. Shao Waae College, The Uivesity of New South Wales, Kesigto,, & KvB Istitute of Techology, Noth Sydey, NSW 00, Austalia Abstact Modula igs ae used to aalyse iteges of the fo N. Whe is odd, the itege stuctue pevets the foatio of pies. Whe is eve, N cooly has a ight-ed-digit of ad so is ot a pie the. Howeve, a sequece defied by q, q0,,,, ca geeate soe pies as the ight-ed-digit is 7. Eleets of this sequece satisfy the o-liea ecuece elatio G G G. eat ubes, whee satisfy this ecuece elatio. Howeve, i this case, the itege stuctue eveals that pies ae liited to <.. Itoductio Aoud 0, eat claied that all ubes of the fo, whee, ae pies []. I 7, Eule showed that whe a coposite is foed []. Hee we shall use the odula igs Z ad Z to aalyse iteges of the fo N (.) i which ay have ay itege value. Whe is odd, the itege stuctue eveals that o pies ca be foed; whe is eve, we get a oe coplex esult. Ou appoach is distictive ad it ca, i a sese, be ustified by ivokig Hoffa s theoy of theoies, aely, a theoy will be accepted by a scietific couity if it explais bette (o oe of) what is kow, fits at its figes with what is kow about othe pats of ou uivese ad akes veifiable, pobably isky, pedictios [].

2 . Modula Rigs These have bee discussed i detail elsewhee [,] so that oly a suay is give hee. Z : Iteges i this ig have the fo ( ), with the class epeseted by i. Eve iteges { 0, }, the latte class havig o powes. Odd iteges {, } i, the latte class havig o eve powes, while iteges i the foe class equal a su of squaes. Z : Iteges i this ig have the fo ( ( i ) ) (Table ). Whe N, iteges { ( eve N), ( odd N) } i, with the class epeseted by i. Thee ae o eve powes i (eve N) o i (odd N). Ay odd N that equals a su of squaes is foud i ( with odd ) ad i (, with eve)., Class Row Table : Z Stuctue The elatioship betwee Z ad Z is suaised i Table. Modula Rig Class Row Class Row Z Z ( ) odd eve ( ) odd o eve Table : Relatioship betwee Z ad Z ( ) eve ( ) odd odd o eve

3 (a) odd.. N as a uctio of ( ) Whe is odd, ( ) ( ) ( ), with, so that N (the ext Class i Table ); hece, N ad N ca eve be pie. (b) eve;. I this case, N. If we coside the ight-ed-digits (REDs, idicated by a asteisk supescipt), the whe, so that ( ) * ( ), * ( ), So that except fo, all such iteges will be coposite. utheoe, ( ) * occus ad N*7 which could idicate a pie o a coposite. I Z, N 0, so that N. I Z, oly Class cotais eve powes (fo eve iteges), ad / N, so that N ad the ow i Table ust be odd. Thus * whe ( ) *. These iteges ca be i eithe of two seies like the eat ubes. The picipal seies has q, but the sub-seies is give by. (.) o exaple, the eat ubes obviously satisfy (.) Seies whee a satisfy X X X. (.) o exaple, fo { : }, { G : }, { H : 0 }, we have a a a ( ) ( ).

4 These types of iteges, whe coposite, have factos of the fo o exaple, fo { },, o, with, G N 8 ( ) ( ( N ( N )( N ). ( 097) ( ( ) )( ( 7777) 9 ) ) (8 ) ( 08 ( )( 0 ) ( 97 79). I geeal the, with two factos, whee N ) (7 ); 9 ) ( N )( N ) ( NN ( N N )) ( NN ( N N ) ) ( ), ( N ). NN N (.) (.) These elatioships will be discussed i oe detail i Sectio. (c) eve;. I the eat seies,, so that. (.7)

5 . acto Stuctue Z so that fo Sectio (b): so that ad Z, ( N )( N ) (.) 7 7 ( )( 79 ) ( 79 09) ( ) ; 8 8 ( 7 )( 87 ) ( ) ( ). O the othe had, suppose ( N )( N ), {,7}. Whe, ( NN N N ) ( NN N) ad so N N N. Obviously, N N, N < ad ae all odd, ad ( N N ), so that the f ( N, N ) caot fit which is, i fact, a pie. I Table, {, ; 9,}. N Soe factos of eat ubes 7 9 ( ) ( 9 )( 9 ) ( 7 )( 97 )( 97 ) ( 79 ) 8 ( 0 ) ( ) ( 9 ) 8 ( ) 7 ( 7 ) Table : Soe factos of eat Nubes 7

6 Whe N Z ( ). Rows of Coposites i Z the ow R is give by [8,9]: t0,,,,,p is the sallest facto of N ad o with ad, as oted above, Whe so that is coposite, ' R R pt. (.) ( ( p ) p), ' R p, ( ( p ) p), ' R p,, ( ), * *, 7. R, ( p ) kp pt, with k o depedig o the class of p. With pm, p beig the sallest facto, t ( M p k ). (.) Obviously, M>(p) o (p) fo positive t. The age of peissible p iceases shaply as iceases (Table ), so that t itege is oe achievable as iceases. Liit of pie age Table : Pie age liits fo factos 8

7 p* k t*,9,7,9,7 7,7 0, 9,7 0, Table : The RED chaacteistics o, the sallest facto is, ad M7007. Sice (x07-), p ad k; thus t 7007 ( ) 9, wheeas, fo, the sallest facto is 777(x9), so p ad k. This yields t Sus of Squaes eat poposed, ad Eule poved, that a pie, p, which satisfies is a su of squaes: p (.) p x y (.) i which (x,y) is uique. This esult follows vey siply fo the itege stuctue. I Z, oly 0 Z cotais powes of eve iteges ad oly Z cotais eve powes of odd iteges. Hece with x odd ad y eve p 0, (.) so that pies ca eve be a su of squaes. The uiqueess of these squae pais has bee show peviously [9]. Coposites ca also have a uique (x,y) pai but i this case x ad y have a coo facto. I Z, the pies appea i both ad, but they ae idetified by the paity of the ows (Table ). Sice {, } (with odd ow) x y. (.) 9

8 Whe (, ) is pie the (x,y) couple is uique as oted above. All iceases, the pobability of fidig a value fo x, othe tha, iceas- ad as es geatly (Table ). have the (x,y) couple (x,y) (,) (,) (,) (,) 9979 (,) (09,) Table : (x,y) pais 7. ial Coets eat s ubes belog to a subgoup whee N, with eve. This subgoup has the lowest iitial values fo (,,,8,) which peits pies to be foed; ( is the state fo aothe goup, (all coposites)). The geeal seies with q (q0,,,, ) is essetially a coposite-foig seies, ad eat s assuptio of pie geeatio is oly tue fo lowest. The factos of the geeal seies have the fo N as show i Table fo the eat ubes. Geeally {, } : it would be of iteest to deteie why these values apply. Coposite status ca be ivestigated fo the chaacteistic ow fuctios fo coposites. All which is equivalet to with odd ow. Hece equals a su of squaes ( x y ) i which x is always uity fo pies; whe is coposite, x has othe values as well, the ube of such values depedig o the ube of factos. (The values of y chage accodigly.) Sice o the factos ay be descibed by fuctios. o exaple, ( ( ) ). ( ( )) ( )( ) (7.) A su ay also epeset, that is,. (7.) 0

9 Accodig to Pepi s Theoe [0], a ecessay ad sufficiet coditio fo to be a pie is that,, ( od ). (7.) This theoe has bee used to test the piality of up to. All values ae coposite whe >. (See Table i [] fo a list of pie factos of eat ubes.) We ote i coclusio though that we ae piaily expoudig aothe theoetical isight ito a old poble i cotast to the oe fashioable coputatioal appoaches. I the wods of Odifeddi [0]: As is ofte the case with techology, ay chages ae fo the wose, ad the atheatical applicatios of the copute ae o exceptio. Such is, fo exaple, the case whe the copute is used as a idiot savat, i the axious ad futile seach fo eve lage pie ubes! Coway ad Guy [] coside eat ubes to the base 0 as well as to the base. They also etio that Gauss poved the supisig fact that if p is a eat pie, the a egula polygo with p sides ca be costucted with ule ad copass, usig Euclid s ules It s said that Gauss equested that a egula 7-go be iscibed o his tobstoe. This was t doe, but thee is a egula 7-go o a ouet to Gauss i Bauschweig (the locatio of the th Iteatioal Reseach Cofeece o iboacci Nubes ad Thei Applicatios i 00). Refeeces. Joh H Coway & Richad K Guy, The Book of Nubes. New Yok: Copeicus, 99.. L. Eule, Opea Oia. Leipzig: Teube, 9.. G.H. Hady & E.M. Wight, A Itoductio to the Theoy of Nubes. d editio. Lodo & New Yok: Oxfod Uivesity Pess, 9.. Roald Hoffa. Why Buy That Theoy? Aeica Scietist, 9() (00): 9-.. J.V. Leyedekkes, J.M. Rybak & A.G. Shao, Itege Class Popeties Associated with a Itege Matix. Notes o Nube Theoy & Discete Matheatics. () (99): -9.. J.V. Leyedekkes, J.M. Rybak & A.G. Shao, Aalysis of Diophatie Popeties Usig Modula Rigs with ou ad Six Classes. Notes o Nube Theoy & Discete Matheatics. () (997): J.V. Leyedekkes, J.M. Rybak & A.G. Shao, The Chaacteistics of Pies ad Othe Iteges withi the Modula Rig Z ad i Class. Notes o Nube Theoy & Discete Matheatics. () (998): J.V. Leyedekkes & A.G. Shao, The Aalysis of Twi Pies withi Z. Notes o Nube Theoy & Discete Matheatics. 7() (00): -.

10 9. J.V. Leyedekkes & A.G. Shao, Usig Itege Stuctue to Cout the Nube of Pies i a Give Iteval, Notes o Nube Theoy & Discete Matheatics. I pess. 0. Piegiogio Odifeddi. The Matheatical Cetuy. (Taslated by Atuo Sagalli; foewod by eea Dyso.) Piceto: Piceto Uivesity Pess, 00.. Has Riesel. Pie Nubes ad Copute Methods fo actoizatio. d editio. Pogess i Matheatics, Volue. Bosto: Bikhäuse, 99. AMS Classificatio Nubes: A, A07